Preface

This third edition marks some 18 years since the second edition of this book appeared and what seems like half a lifetime ago—some 31 years—since the first edition was written. It has been extremely gratifying that the book has lasted this long, that it continues to be used by many and that a new edition was welcomed by Wiley.

Since the second edition the subject has consolidated and largely turned to more and more areas of application, including a renewed interest from the geotechnical engineering research community. But also in practice structural reliability increasingly is being applied, particularly for situations where quantitative, data-based risk assessment of non-elementary structural or other systems is required. Overviews of the papers contributed to conferences such as ICASP, ICOSSAR, IFIP, IALCCE and CSM shows much attention paid to applications and relatively little to sorting out some of the remaining really challenging theoretical problems such as how to deal with complex systems with a multitude of random variables or processes, and for which many potential failure modes and combination of such modes may exist. Fortunately, the availability of ever greater computational power has meant that enumeration methods, once thought to be the way forward for dealing with really complex problems, can be cast aside in favour of sheer brute force number crunching. In this sense Chapter 3 and the parts of Chapter 5 dealing with Monte Carlo methods are now more important, for practical problems, than the elegant but simpler FOSM/FOR/SOR methods that allow easier insight into ‘what was driving what’.

The present edition follows much of the second edition but updates areas such as Monte Carlo methods, systems reliability, some aspects of load and resistance modelling, code calibration, analysis of existing structures and adds, for the first time, a chapter on optimization in the context of structural reliability. The co-author for this edition, André T. Beck, has contributed much to these changes, as well as to the worked examples provided where relevant for each chapter and collected together in Appendix F. We have had the good fortune to have at hand the many comments and corrections, principally supplied by Dr. Bill Gray during his post-doctoral days at The University of Newcastle. As before, we have had to be selective in our coverage and have had to make difficult decisions about what to include and what leave out.

Now, as 18 years ago, a surf or a beach run at Newcastle's wonderful Pacific Ocean beaches, a surf or a bike ride along the south-eastern Brazilian coast, seem better ways to spend one's time than revising a book. Our spouses tell us so, our colleagues tell us so, our minds tell us so, but what do we do?

10 February 2017

Robert E. Melchers

Bar Beach, Newcastle

André T. Beck

Florianópolis, SC

Preface to the Second Edition

It is over ten years since the first edition of this book appeared and more than 12 years since the text was written. At the time structural reliability as a discipline was evolving rapidly but was also approaching a degree of maturity. Perhaps it is not surprising, then, that rather little of the first edition now seems out-dated.

This edition differs from the first mainly in matters of detail. The overall layout has been retained but all of the original text has been reviewed. Many sections have been partly rewritten to make them clearer and more complete and many, often small but annoying, errors and mistakes have been corrected. Hopefully not too many new ones have crept in. Many new references have been added and older, now less relevant, ones deleted. This is particularly the case in referring to applications, in which area there has been much progress.

The most significant changes in this edition include the up-dating of the sections dealing with Monte Carlo simulation, the addition of the Nataf transformation in the discussion of FOSM/FORM methods, some comments about asymptotic methods, additional discussion of structural systems subject to multiple loads and a new chapter devoted to the safety checking of existing structures, an area of increasing importance.

Other areas in which there have been rapid developments, such as simulation of random processes and random fields, applications in structural dynamics and fatigue and specialist refinements of theory are all of interest but beyond the scope of an introductory book. Readers might care to refer to the specialist literature, proceedings of conferences such as the ICASP, ICOSSAR and IFIP series and to journals such as Structural Safety, Probabilistic Engineering Mechanics and the Journals of Engineering Mechanics and Structural Engineering of ASCE. Overviews of various aspects of applications of structural reliability are given also in Progress in Structural Engineering and Mechanics. There are, of course, other places to look, but these should form a good starting point for keeping in touch with theoretical developments and applications.

In preparing this edition I had the good furtune to have at hand a range of comments, notes and advice. I am particularly indebted to my immediate colleagues Mark Stewart and Dimitry Val for their critical comments and their assistance with some of the new sections. Former research students have also contributed and I mention in this regard particularly H.Y. Chan, M. Moarefzadeh and X.L. Guan. Naturally, I owe a very significant debt to the international structural reliability community in general and to some key people in particular, including Ove Ditlevsen, Rudiger Rackwitz, Armen Der Kiureghian and Bruce Ellingwood—they, and many others, will know that I appreciate their forebearance and friendship.

The encouragement and generous comments from many sources is deeply appreciated. It has contributed to making the hard slog of revision a little less painful. Sometimes a beach run or a surf seemed a better alternative to spending an hour or so making more corrections to the text… As before, the forebearance of my family is deeply appreciated. Like many academic households they have learnt that academics are their own worst enemies and need occasionally to be dragged away from their Macintoshes to more socially acceptable activities.

August, 1998

Robert E. Melchers

Bar Beach, Newcastle

Preface to the First Edition

The aim of this book is to present a unified view of the techniques and theory for the analysis and prediction of the reliability of structures using probability theory. By reliability, in this context, will be understood not just reliability against extreme events such as structural collapse or facture, but against the violation of any structural engineering requirements which the structure is expected to satisfy.

In practice, two classes of problems may arise. In the first, the reliability of an existing structure at the ‘present time’ is required to be assessed. In the second, and much more difficult class, the likely reliability of some future, or as yet uncompleted, structure must be predicted. One common example of such a requirement is in structural design codes, which are essentially instruments for the prediction of structural safety and serviceability supported by previous experience and expert opinion. Another example is the reliability assessment of major structures such as large towers, offshore platforms and industrial or nuclear plants for which structural design codes are either not available or not wholly acceptable. In this situation, the prediction of safety both in absolute terms and in terms of its interrelation to project economics is becoming increasingly important. This class of assessment relies on the (usually reasonable but potentially dangerous) assumption that past experience can be extrapolated into the future.

It might be evident from these remarks that the analysis (and prediction) of structural reliability is rather different from the types of analysis normally performed in structural engineering. Concern is less with details of stress calculations, or member behaviour, but rather with the uncertainties in such behaviour and how this interacts with uncertainties in loading and in material strength. Because such uncertainties cannot be directly observed for any one particular structure, there is a much greater level of abstraction and conceptualization in reliability analysis than is conventionally the case for structural analysis or design. Modelling is not only concerned with the proper and appropriate representation of the physics of any structural engineering problem, but also with the need to obtain realistic, sufficiently simple and workable models or representations of both the loads and the material strengths, and also their respective uncertainties. How such modelling might be done and how such models can be used to analyse or predict structural reliability is the central theme of this book.

In one important sense, however, the subject matter has a distinct parallel with conventional structural engineering analysis and its continual refinement; that is, that ultimately concern is with costs. Such costs include not only those of design, construction, supervision and maintenance but also the possible cost of failure (or loss of serviceability). This theme, although not explicitly pursued throughout the book, is nevertheless a central one, as will become clear in Chapter 2. The assessment or predictions obtained using the methods outlined in this book have direct application in decision-making techniques such as cost-benefit analysis or, more precisely when probability is included, risk-benefit analysis. As will be seen in Chapter 9, one important area of application for the methods presented here is in structural design codes, which, it will be recognized, are essentially particular (if perhaps rather crude and intuitive) forms of risk-benefit methodology.

A number of other recent books have been devoted to the structural reliability theme. This book is distinct from the others in that it has evolved from a short course of lectures for undergraduate students as well as a 30-h graduate course of lectures which the author has given periodically to (mainly) practising structural engineers during the last 8 years. It is also different in that it does not attempt to deal with related topics such as spectral analysis for which excellent introductory texts are already available.

Other features of the present book are its treatment of structural system reliability (Chapter 5) and the discussion of both simulation methods (Chapter 3) and modern second-moment and transformation methods (Chapter 4). Also considered is the important topic of human error and human intervention in the relationship between calculated (or ‘nominal’) failure probabilities and those observed in populations of real structures (Chapter 2).

The book commences (Chapter 1) by reviewing traditional methods of defining structural safety such as the ‘factor of safety’, the ‘load factor’, ‘partial factor’ formats (i.e. ‘limit state design’ formats) and the ‘return period’. Some consistency aspects of these methods are then presented and their limited use of available data noted, before a simple probabilistic safety measure, the ‘safety margin’ and the associated failure probability are introduced. This simple one-load one-resistance model is sufficient to introduce the fundamental ideas of structural reliability assessment. Apart from Chapter 2, the rest of the book is concerned with elaborating and illustrating the reliability analysis and prediction theme.

While Chapters 3, 4 and 5 deal with particular calculation techniques for time-independent situations, Chapter 6 is concerned with extending the ‘return period’ concept introduced in Chapter 1 to more general formulations for time-dependent problems. The three principal methods for introducing time, the time-integrated approach, the discrete time approach and the fully time-dependent approach, are each outlined and examples given. The last approach is considerably more demanding than the other two (classical) methods since it is necessary to introduce elements of stochastic process theory. First-time readers may well decide to skip rather quickly through much of this chapter. Applications to fatigue problems and structural vibrations are briefly discussed from the point of view of probability theory, but again the physics of these problems is outside the scope of the present book.

Modelling of wind and floor loadings is described in Chapter 7 whilst Chapter 8 reviews probability models generally accepted for steel properties. Both load and strength models are then used in Chapter 9. This deals with the theory of structural design codes and code calibration, an important area of application for probabilistic reliability prediction methodology.

It will be assumed throughout that the reader is familiar with modern methods of structural analysis and that he (or she) has a basic background in statistics and probability. Statistical data analysis is well described in existing texts; a summary of probability theory used is given in Appendix A for convenience.

Further, reasonable competence in applied mathematics is assumed since no meaningful discussion of structural reliability theory can be had without it. The level of presentation, however, should not be beyond the grasp of final-year undergraduate students in engineering. Nevertheless, particularly difficult theoretical sections which might be skipped on a first reading are marked with an asterisk (*).

For teaching purposes, Chapters 1 and 2 could form the basis for a short undergraduate course in structural safety. A graduate course could take up the topics covered in all chapters, with instructors having a bias for second-moment methods skipping over some of the sections in Chapter 3 while those who might wish to concentrate on simulation could spend less time on Chapter 4. For an emphasis on code writing, Chapters 3 and 5 could be deleted and Chapters 4 and 6 cut short.

In all cases it is essential, in the author's view, that the theoretical material be supplemented by examples from experience. One way of achieving this is to discuss particular cases of structural failure in quite some detail, so that students realize that the theory is only one (and perhaps the least important) aspect of structural reliability. Structural reliability assessment is not a substitute for other methods of thinking about safety, nor is it necessarily any better; properly used, however, it has the potential to clarify and expose the issues of importance.

Acknowledgements

This book has been a long time in the making. Throughout I have had the support and encouragement of Noel Murray, who first started me thinking seriously about structural safety, and also of Paul Grundy and Alan Holgate. In more recent times, research students Michael Harrington, Tang Liing Kiong, Mark Stewart and Chan Hon Ying have played an important part.

The first (and now unrecognizable) draft of part of the present book was commenced shortly after I visited the Technical University, Munich, during 1980 as a von Humboldt Fellow. I am deeply indebted to Gerhart Schueller, now of Universität Innsbruck, for arranging this visit, for his kind hospitality and his encouragement. During this time, and later, I was also able to have fruitful discussions with Rudiger Rackwitz.

Part of the last major revision of the book was written in the period November 1984-May 1985, when I visited the Imperial College of Science and Technology, London, with the support of the Science and Engineering Research Council. Working with Michael Baker was a most stimulating experience. His own book (with Thoft-Christensen) has been a valuable source of reference.

Throughout I have been extremely fortunate in having Mrs. Joy Helm and more recently Mrs. Anna Teneketzis turn my difficult manuscript into legible typescript. Their cheerful co-operation is very much appreciated, as is the efficient manner with which Rob Alexander produced the line drawings.

Finally the forbearance of my family was important, many a writing session being abruptly concluded with a cheerful ‘How's Chapter 6 going, Dad?’

December 1985

Robert E. Melchers

Monash University

Chapter 1

Measures of Structural Reliability

1.1 Introduction

The manner in which an engineering structure will respond to loading depends on the type and magnitude of the applied load and the structural strength and stiffness. Whether the response is considered satisfactory depends on the requirements that must be satisfied. These include safety of the structure against collapse, limitations on damage, or on deflections or other criteria. Each such requirement may be termed a limit state. The ‘violation’ of a limit state can then be defined as the attainment of an undesirable condition for the structure. Some typical limit states are given in Table 1.1.

Table 1.1 Typical limit states for structures

From observation it is known that very few structures collapse, or require major repairs, etc., so that the violation of the most serious limit states is a relatively rare occurrence. When violation of a limit state does occur, the consequences may be extreme, as exemplified by the spectacular collapses of structures such as the Tay Bridge (wind loading), Ronan Point Flats (gas explosion), Kielland Offshore Platform (local strength problems), Kobe earthquake (ductility), etc.

The study of structural reliability is concerned with the calculation and prediction of the probability of limit state violation for an engineered structural system at any stage during its life. In particular, the study of structural safety is concerned with the violation of the ultimate or safety limit states for the structure. More generally, the study of structural reliability is concerned with the violation of performance measures (of which ultimate or safety limit states are a subset). This broader definition allows the scope of application to move from structural criteria as specified in traditional design codes (Chapter 9) to broader-based performance requirements for structures, such as might be used in design optimization processes (Chapter 11).

In the simplest case, the probability of occurrence of an event such as limit state violation is a numerical measure of the chance of its occurrence. This measure either may be obtained from measurements of the long-term frequency of occurrence of the event for generally similar structures, or it may be simply a subjective estimate of the numerical value. In practice it is seldom possible to observe for a sufficiently long period of time, and a combination of subjective estimates and frequency observations for structural components and properties may be used to predict the probability of limit state violation for the structure.

In probabilistic assessments any uncertainty about a variable (expressed, as will be seen, in terms of its probability density function) is taken into account explicitly. This is not the case in traditional ways of measuring safety, such as the ‘factor of safety’ or ‘load factor’. These are ‘deterministic’ measures, since the variables describing the structure, its strength and the applied loads are assumed to take on known (if conservative) values about which there is assumed to be no uncertainty. Precisely because of their traditional and really quite central position in structural engineering, it is appropriate to review the deterministic safety measures prior to developing probabilistic safety measures.

1.2 Deterministic Measures of Limit State Violation

1.2.1 Factor of Safety

The traditional method to define structural safety is through a ‘factor of safety’, usually associated with elastic stress analysis and which requires that:

1.1 equation

where σi(ϵ) is the i th applied stress component calculated to act at the generic point ϵ in the structure, and σpi is the permissible stress for the i th stress component.

The permissible stresses σpi are usually defined in structural design codes. They are derived from material strengths (ultimate moment, yield point moment, squash load, etc.), expressed in stress terms σui but reduced through a factor F:

1.2 equation

where F is the ‘factor of safety’. The factor F may be selected on the basis of experimental observations, previous practical experience, economic and, perhaps, political considerations. Usually, its selection is the responsibility of a code committee.

According to (1.1), failure of the structure should occur when any stressed part of it reaches the local permissible stress. Whether failure actually does occur depends entirely on how well σi(ϵ) represents the actual stress in the real structure at ϵ and how well σpi represents actual material failure. It is well known that observed stresses do not always correspond well to the stresses calculated by linear elastic structural analysis (as commonly used in design). Stress redistribution, stress concentration and changes due to boundary effects and the physical size effect of members all contribute to the discrepancies.

Similarly, the permissible stresses that, commonly, are associated with linear elastic stress analysis are not infrequently obtained by linear scaling down, from well beyond the linear region, of the ultimate strengths obtained from tests. From the point of view of structural safety, this does not matter very much, provided that the designer recognizes that his calculations may well be quite fictitious and provided that (1.1) is a conservative safety measure.

By combining expressions (1.1) and (1.2) the condition of ‘limit state violation’ can be written as

1.3

Expressions (1.3) are ‘limit state equations’ when the inequality sign is replaced by an equality. These equations can be given also in terms of stress resultants, obtained by appropriate integration:

1.4

where Ri is the i th resistance at location ϵ and Si is the i th stress resultant (internal action). In general, the stress resultant Si are made up of the effects of one or more applied loads Qj; typically

where D is the dead load, L is the live load and W is the wind load.

The term ‘safety factor’ also has been used in another sense, namely in relation to overturning, sliding, etc., of structures as a whole, or as in geomechanics (dam failure, embankment slip, etc.). In this application, expressions (1.3) are still valid provided that the stresses σui and σi are interpreted appropriately.

1.2.2 Load Factor

The ‘load factor’ λ is a special kind of safety factor developed for use originally in the plastic theory of structures. It is the theoretical factor by which a set of loads acting on the structure must be multiplied, just enough to cause the structure to collapse. Commonly, the loads are taken as those acting on the structure during service load conditions. The strength of the structure is determined from the idealized plastic material strength properties for structural members [Heyman, 1971].

For a given collapse mode (i.e. for a given ultimate ‘limit state’), the structure is considered to have ‘failed’ or collapsed when the plastic resistances Rpi are related to the factored loads λQj by

1.5 equation

where RP is the vector of all plastic resistances (e.g. plastic moments) and Q is the vector of all applied loads. Also, WR( ) is the internal work function and WQ( ) the external work function, both described by the plastic collapse mode being considered.

If proportional loading is assumed, as is usual, the load factor can be taken out of parentheses. Also the loads Qj usually consist of several components, such as dead, live, wind, etc. Thus (1.5) may be written in the form of a limit state equation:

with ‘failure’ denoted by the left-hand side being less than unity.

Clearly there is much similarity in formulation between the factor of safety and the load factor as measures of structural safety. What is different is the reference level at which the two measures operate: the first at the level of working loads and at the ‘member’ level; the second at the level of collapse loads and at the ‘structure level’.

1.2.3 Partial Factor (‘Limit State Design’)

A development of the above two measures of safety is the so-called ‘partial factor’ approach. For limit state i it can be expressed at the level of stress resultants (i.e. member design level) as

1.6 equation

where R is the member resistance, φ is the partial factor on R and SD, SL are the dead and live load effects respectively with associated partial factors γD, γL. Expression (1.6) was originally developed during the 1960s for reinforced concrete codes. It enabled the live and wind loads to have greater ‘partial’ factors than the dead load, in view of the former's greater uncertainty, and it allowed a measure of workmanship variability and uncertainty about resistance modelling to be associated with the resistance R [MacGregor, 1976]. This extension of earlier safety formats had considerable appeal since it allowed better representation of the factors and uncertainties associated with loadings and resistances.

For a plastic collapse analysis at the structure level, formulation (1.6) becomes

where R and Q are vectors of resistance and loads respectively. Clearly the partial factors (φ, γ) in this expression will be different from those of expression (1.6).

Figure 1.1 Bending moment diagrams for Example 1.1.

Example 1.1

The simple portal frame of Figure 1.1(a) is subject to loads Q1 and Q2. If the relative moments of inertia of the members are known, the elastic bending moment diagram can be found as in Fig 1.1(b). The ‘limit states’ for bending capacity are then

where φ, γ1 and γ2 are partial factors described by a structural design code. The MCi are the ultimate moment capacities required at sections c01-math-0007 for the structure to be considered ‘just safe’.

If the frame is to be designed or analysed assuming rigid-plastic theory, the relative distribution of the plastic moments Mpi around the frame must be known or assumed. If they all are equal, the plastic bending moment diagram of [Figure 1.1(c) is obtained and only one limit state equation is needed for sections 1–3:]

where now Mpi is the required plastic moment capacity at sections 1, 2 and 3 and where φp, γp1 and γp2 are now code-prescribed partial factors for plastic structural systems.

1.2.4 A Deficiency in Some Safety Measures: Lack of Invariance

From Example 1.1, it will be evident that the partial factors φ and c01-math-0008 in (1.6) depend on the limit state being considered. Hence they depend on the definitions of R, SD and SL. However, even for a given limit state, these definitions are not necessarily unique, and therefore the partial factors may not be unique either. This phenomenon is termed the ‘lack of invariance’ of the safety measure. It arises because there are different ways in which the relationships between resistances and loads may be defined. Some examples of this are given below. Ideally, the safety measure should not depend on the way in which the loads and resistances are defined.

Figure 1.2 Example 1.2: Structure subject to overturning under lateral load H and with vertical load W and supported by two columns applying vertical forces V1 and V2.

Example 1.2

The structure shown in Figure 1.2 is supported on two columns. The capacity of column B is c01-math-0009 in compression. The safety of the structure can be measured in three different ways using the traditional ‘factor of safety’ F:

a.Overturning resistance about A

b.Capacity of column B

c.Net capacity of column B (resistance minus load effect of W)

All three of these factors of safety c01-math-0010 for column B apply to the same structure and the same loading, so that the difference in the values of Fi is due entirely what is considered to represent the resistance of the structure and what is considered to be the applied load. In general such a difference in outcomes is not helpful for the unique definition of a factor of safety. However, for some special cases of the partial factors the outcomes can be made the same. Thus it is easily verified that the calculations give the identical result c01-math-0011 if a partial safety factor c01-math-0012 is applied to the resistance R, thus:

Similarly, the result c01-math-0013 would be achieved if the loads H and W were factored by c01-math-0014 . More generally, any choice of combination of φ and γ could be appropriate, provided that c01-math-0015 . This can be expressed as:

A different way of defining a measure of safety is the ‘safety margin’. It measures the excess of resistance compared with the stress resultant (or loading); thus:

1.7 equation

For the present example, the safety margins are

1.7a equation

1.7b equation

and

1.7c equation

It is readily verified that when c01-math-0020 , i.e. at the point of failure, these three expressions are equivalent. This shows that the safety margin concept of safety is ‘invariant’ with respect to the limit state functions (1.7a–c).

Example 1.3 [adapted from Ditlevsen, 1973]

The reinforced concrete beam shown in Figure 1.3 (a) has a moment capacity R when it is subject to an axial force N and a moment M applied at the beam centroid c01-math-0021 . Both N and M are composed of the effects of a dead load and a live load: c01-math-0022 and c01-math-0023 . The moment capacity calculated about c01-math-0024 is c01-math-0025 , from simple statics. (Note that the actual moment capacity of the beam is not changed!) Also, at c01-math-0026 , the applied moment is given by c01-math-0027 . The state ‘just safe’ can now be defined for given moment capacity R, and given axial force N, by the factor of safety as:

1.8a equation

1.8b

In this format c01-math-0030 is true only when c01-math-0031 . This means that the factor of safety depends on the convention chosen for the origin of the applied actions and of the resistance. If, as in Example 1.2, R is replaced by the factored term φR, such that c01-math-0032 , then it follows readily that F1 is also unity. Hence, provided that ‘partial factor’ φ is chosen in such a way that the ‘factor of safety’ F is unity, the origin chosen to define R, N and M is immaterial. A similar result holds if N and M are replaced by γ N and γ M, where γ is an appropriately chosen partial factor on the loading.

Figure 1.3 Reinforced concrete beam: Example 1.3.

The state ‘just safe’ can be written also in the partial factor format of (1.6). Indeed, noting that c01-math-0033 and c01-math-0034 , at c01-math-0035 it follows that

1.9a equation

and at c01-math-0037 , treating, as before, c01-math-0038 as the resistance to bending,

1.9b

Subtracting (1.9a) from (1.9b) and dividing out by a leaves

1.10 equation

Since in general c01-math-0041 , it follows that (1.10) will be satisfied only if either c01-math-0042 or c01-math-0043 . Except for c01-math-0044 these expressions are both inconsistent with the conventional interpretation that c01-math-0045 (to reduce the calculated resistance) and c01-math-0046 (to increase the loads or applied stresses).

The reason for this result should be clear. In (1.14) the term c01-math-0047 on the left-hand side was treated as a resistance, per se, whereas it is strictly a resistance effect caused directly by the applied loading (note that it is not affected by workmanship, material strength, etc., as is R). The key to an invariant safety measure is thus at hand. Partial factors such as φ should be applied directly to resistances only, and partial factors such as γ to loads only, and the direct application of (1.6) to a mixed variable c01-math-0048 is not correct.

It is important to note that the safety margin Z (Equation 1.7) is invariant for both definitions of resistance in this example. In the first case c01-math-0049 , while in the second case

.

1.2.5 Invariant Safety Measures

As can be seen from the above examples, one form of invariant safety measure is obtained if the resistances Ri and the loads Qj acting on the structure are so factored that the ratio between any relevant pair φiRi and γjQj is unity at the point of limit state violation. In simple terms, this requires that all variables be reduced to a common base before being compared. This is the case for the permissible stress measure of structural safety expressed by equation (1.3). Another and important form of invariant safety measure is the safety margin c01-math-0051 defined in equation (1.7). It will be used extensively in the sections to follow because of its invariant properties.

Some readers may recognize a parallel between the above discussion and the decision criteria in cost-benefit analysis. The safety margin corresponds to the ‘net present value’ criterion and the problem of safety factor invariance to the ‘numerator-denominator’ problem [e.g. Prest and Turvey, 1965].

1.3 A Partial Probabilistic Safety Measure of Limit State Violation—The Return Period

In the historical development of engineering design, loads due to natural phenomena such as winds, waves, storms, floods, earthquakes, etc. were recognized quite early as having randomness in time as well as in space. The randomness in time was considered in terms of the ‘return period’. The return period is defined as the average (or expected) time between two successive statistically independent events. Of course, the actual time T between events is a random variable.

In most practical applications an ‘event’ constitutes the exceedance of a certain threshold, for example as associated with loading (e.g. wind velocity c01-math-0052 ). Such an event may be used to define a ‘design load’ and the design of the structure itself is then usually considered deterministically, i.e. using conventional design procedures. Hence this approach is only a partially probabilistic method.

The return period may be defined as follows. For independent samples from a population (i.e. for a Bernoulli trial sequence), the trial T on which the first occurrence of an event takes place is given by the geometric distribution (A.23), which states that the probability that the first occurrence occurs on the t th trial is:

1.11

where p is the probability of occurrence of the event (e.g. c01-math-0054 ) in any one trial and c01-math-0055 is the probability that the event does not occur. If trials are now interpreted as time intervals, during each of which only the occurrence of events c01-math-0056 is recorded, the first occurrence of an event becomes the ‘first occurrence time’, given by expression (1.11). The ‘mean recurrence time’ or the ‘return period’ is then the expected value of T (see A. 10):

1.12

where c01-math-0058 is the cumulative distribution function of X.

Thus the return period c01-math-0059 is equal to the reciprocal of the probability of the occurrence of the event in any one (or a single) time interval. For most engineering problems, the chosen time interval is one year, so that p is the probability of occurrence of the event c01-math-0060 in any one year (e.g. the probability that a load c01-math-0061 will occur (at least once) during the year). Then c01-math-0062 is the number of years, on average, between events.

Because the exceedance events that occur during a time period (e.g. during a year) are associated with the end of that period, c01-math-0063 is dependent on the time period chosen [Borgman, 1963]. This is illustrated in Figure 1.4, where four exceedance events, A, B, C and D are shown occurring after an arbitrary initial event 0. The mean recurrence time c01-math-0064 for the actual observations is shown in Figure 1.4(a) and is given by the average of the distance (i.e. time) between the events, i.e. by c01-math-0065 years.

Figure 1.4 Idealizations of actual load phenomenon for the ‘return period’ concept.

In Figure 1.4(b) with the time period taken as 1 year, and the events counted at the end of each time period, it follows easily that c01-math-0066 years. Similarly, for c01-math-0067 years. However, when a 4-year time period is used (Figure 1.4(d)) two of the events in each period are counted as one at the end of the period, and c01-math-0068 in this case becomes 4 years.

This somewhat artificial example shows three things. Firstly, that the return period depends, as noted, on the definition of the time scale, and secondly that the possible occurrence of more than one event within a time period is ignored. This means that, where event occurrence is relatively frequent compared with the time period employed, the return period measure is not accurate.

The third and a most important point is that the probability distribution of the magnitude of X (i.e. the phenomenon being considered) is not considered. Only magnitudes c01-math-0069 are counted. This means that the return period is a probabilistic measure in terms of time only, but not in terms of the magnitude of the loading and its interaction with the resistance.

It should be clear that in practice the events may not be independent, as postulated, particularly if the events occur rather frequently. Fortunately, the return period concept is used mainly for rather rare events (i.e. the level X is quite high), and it is then reasonable to assume event independence. Time scale dependence is then also not a significant issue. Chapter 6 gives a much more detailed discussion.

Example 1.4

For a structure subject to a ‘50-year wind’ of 60 km/h velocity:

a. the return period for a 60-km/h c01-math-0070 years

b. the probability of exceeding 60 km/h in any one year is

c. the probability of exceeding the design wind velocity (i.e. c01-math-0071 ) for the first time during the fourth year, is (geometric distribution A.23):

d. the probability of exceeding the design wind velocity in only one of the years in a 4-year period is given by the binomial distribution (A.17):

e.

the probability of exceeding the design wind velocity (i.e. c01-math-0072 ) during any of the years in a 4 year period is given by the geometric distribution (A.23):

or alternatively,

Note that the period 4 years can be generalized to ‘design life’ tL and the question rephrased to ‘the probability of exceeding the design velocity within the design life’:

1.13

Some typical values for the relationship between the exceedance probability c01-math-0074 the return period c01-math-0075 and the design life tL are given in Table 1.2 [Borgman, 1963].

Table 1.2 Return period c01-math-0076 as function of design life tL and exceedance probability PT (T ≤ tL).

f. the probability of exceeding the design wind velocity within the return period is

but

where c01-math-0080 . Hence

Note that even for smaller c01-math-0081 , this result is a good approximation; thus, for c01-math-0082 ,

This shows that there is a chance of about 2 in 3 that the exceedance event will occur within a design life equal to the return period.

1.4 Probabilistic Measure of Limit State Violation

1.4.1 Introduction

The return period concept considers only the probability that a loading exceeds a set limit and assumes such exceedances (or ‘level crossings’ – see Chapter 6) to be randomly distributed in time. This is a useful improvement over deterministic descriptions of loading but ignores the fact that, even at a given point in time, the actual value of the loading is uncertain. This is illustrated in Figure 1.5 for floor loading.

Figure 1.5 Histogram of private office live loads [after Culver, 1976].

The histogram of Figure 1.5 shows, for example, that the probability that the floor loading lies between 0.6 and 0.7 kPa is about 7%. Such information is obtained from actual surveys of floor loads (see Chapter 7), and can be represented by the probability density function fQ(q). (Recall that fQ( ) denotes the probability that the load Q will take on a value between q and c01-math-0083 as c01-math-0084 - see also Section A.3.) The load Q can be converted to a load effect S by conventional structural analysis procedures. Using the same transformation(s), the probability density function fS( ) can be obtained also, if necessary, using methods such as outlined in Section A.10. However, details of this need not be of concern for the present.

Resistance, geometric and workmanship variables and many others may be described similarly in probabilistic terms. For example, a typical resistance histogram and the inferred probability distribution for the yield strength of steel are shown in Figure 1.6. Naturally, material strengths such as steel yield strength can be converted to member resistance R by multiplying by section properties (such as A, the cross-sectional area). Then it is possible to determine a probability density function fR( ).

Figure 1.6 Histogram and inferred distribution for structural steel yield strength [adapted from Alpsten, 1972 with permission of ASCE].

In general, the loads applied to a structure fluctuate with time and are of uncertain value at any one point in time. This is carried over directly to the load effects (or internal actions) S. Somewhat similarly the structural resistance R will be a function of time (but not usually a fluctuating one) owing to fatigue, deterioration and similar actions. Loads have a tendency to increase, and resistances to decrease, with time. It is likely also that the uncertainty in both quantities increases with time, particularly if they have to be predicted. This means that the probability density functions fS( ) and fR( ) become wider and flatter with time and that the mean values of S and R also change with time. As a result, the general reliability problem can be represented as in Figure 1.7.

Figure 1.7 Schematic time-dependent reliability problem.

The safety limit state will be violated whenever, at any time t,

1.14

The probability that this occurs for any one (single) load application (or load cycle) is the probability of limit state violation, or simply the probability of failure pf. Roughly, it may be represented by, but is not actually equal to, the amount of overlap of the probability density functions fR and fS in Figure 1.7. Since this overlap may vary with time, pf also may be a function of time.

To make the problem more tractable, it is convenient for many situations to assume that Q and R are ‘time-invariant’, that is they are not functions of time. An example of this is the case when the load Q is applied to the structure only once and the probability of limit state violation is sought for that particular load application only.

However, if the load is applied many times (e.g. a single time-varying load might be considered this way) and R is taken as constant, then the maximum value of that load (within a given time interval [0, T]) is of interest if it is assumed that the structure will fail under the (once-only) application of this maximum load. One way to properly represent this maximum load is through the use of an extreme value distribution, such as the Gumbel (EV-I) or Frechet (EV-II) distributions (see Appendix A). If this is done, the effect of time may be ignored in the reliability calculations. This approach is not satisfactory when more than one load is involved or when the resistance changes with time. Discussion of these matters and the more general reliability problem is deferred to Chapter 6.

1.4.2 The Basic Reliability Problem

The basic structural reliability problem considers only one load effect S resisted by one resistance R. Each is described by a known probability density function, fS( ) and fR( ) respectively. As noted, S may be obtained from the applied loading Q through a structural analysis (either deterministic or with random components). It is important that R and S be expressed in the same units.

For convenience, but without loss of generality, only the safety of a structural element will be considered here and, as usual, that structural element will be considered to have failed if its resistance R is less than the stress resultant S acting on it. The probability pf of failure of the structural element can then be stated in any of the following ways:

1.15a equation

1.15b equation

1.15c equation

1.15d equation

or in general

1.15e equation

where G( ) is termed the ‘limit state function’ and the probability of failure is identical with the probability of limit state violation. Equations (1.15) could, of course, also have been written in terms of R and Q for the structure as a whole.

Quite general (marginal) density functions fR and fS for R and S respectively are shown in Figure 1.8 together with the joint (bivariate) density function fRS(r, s) (see also Section A.6). For any infinitesimal element (Δr Δs), the latter represents the probability that R takes on a value between r and c01-math-0091 and S a value between s and c01-math-0092 as Δr and Δs each approach zero. In Figure 1.8, Equations (1.15) are represented by the hatched failure domain D, so that the failure probability may be written as:

1.16

When R and S are independent, c01-math-0094 (see A.6.3), and (1.16) becomes:

1.17

Noting that for any random variable X, the cumulative distribution function is given by (A.8):

provided c01-math-0096 , it follows that for the common, but special, case when R and S are independent, (1.17) can be written in the single integral form:

1.18

This is also known as a ‘convolution integral’ with meaning easily explained by reference to Figure 1.9. FR(x) is the probability that c01-math-0098 or the probability that the actual resistance R of the member is less than some value x. This represents failure if the loading is c01-math-0099 . The probability that this is the case is given by the term fS(x) that represents the probability that the load effect S acting in the member has a value between x and c01-math-100 in the limit as c01-math-101 . By considering all possible values of x, i.e. by taking the integral over all x, the total failure probability is obtained. This is also seen in Figure 1.10 where the (marginal) density functions fR and fS have been drawn along the same axis.

Figure 1.8 Space of the two random variable (r, s) and the joint density function fRS(r, s), the marginal density functions fR and fS and the failure domain D.

Figure 1.9 Basic c01-math-102 problem: FR( )fS( ) representation.

Figure 1.10 Basic c01-math-103 problem: fR( ) fS( ) representation.

Through integration of fR( ) in (1.17), the order of integration was reduced by one. This is convenient and useful, but not general. It was only possible because R was assumed independent of S. In general, dependence between variables should be considered. This more complex situation is discussed further is Section 1.5 and Chapters 3 and 4.

For the present, restricting attention to simpler formulations, an alternative to expression (1.18) is:

1.19 equation

This can be seen to be simply the ‘sum’ of the failure probabilities over all the cases of resistance for which the load exceeds the resistance.

The lower limit of integration shown in Expressions (1.17) to (1.19) may not be totally satisfactory, since a ‘negative’ resistance usually is physically not possible. The lower integration limit therefore strictly should be zero, although this may be inconvenient and slightly inaccurate if R or S or both are modelled by distributions unlimited in the lower tail (such as the Normal or Gaussian distribution). The inaccuracy arises strictly from the modelling of R and/or S, and not from the theory involved with (1.17) to (1.19). This important point is sometimes overlooked in discussions about appropriate distributions to represent random variables.

1.4.3 Special Case: Normal Random Variables

For a few distributions of R and S it is possible to integrate the convolution integral (1.18) analytically. The most notable example is when both R and S are normal random variables with means μR and μS and variances c01-math-105 and c01-math-106 respectively. The safety margin c01-math-107 then has a mean and variance given by well-known rules for addition (subtraction) of normal random variables:

1.20a equation

1.20b equation

Equation (1.15b) then becomes

1.21

where Φ( ) is the standard normal distribution function (zero mean and unit variance) extensively tabulated in statistics texts (see also Appendix D). The random variable c01-math-111 is shown in Figure 1.11, in which the failure region c01-math-112 is shown shaded. Using (1.20) and (1.21) it follows that [Cornell, 1969a]

1.22 equation

where c01-math-114 is defined as the ‘safety index’ (1.21).

Figure 1.11 Distribution of safety margin c01-math-115 .

If either of the standard deviations σS or σR or both is increased, the term in square brackets in (1.22), will become smaller and hence pf will increase, as might be expected. Similarly if the difference between the mean of the load effect and the mean of the resistance is reduced, pf increases. These observations may be deduced also from Figure 1.7, taking the amount of overlap of fR( ) and fS( ) as a rough indicator of pf at any point in time.

Example 1.5

A simply supported timber beam of length 5 m is loaded with a central load Q having mean c01-math-116 and variance c01-math-117 . The bending strength of similar beams has been found to have a mean strength c01-math-118 with a coefficient of variation (COV) of 0.15. It is desired to evaluate the probability of failure.

Assume that the beam self-weight and any variation in the length of the beam can be ignored. From basic structural theory, the applied moment (the load effect S) at the centre of the beam (due to the load Q) is given by c01-math-119 . Since c01-math-120 it follows that the mean load effect and the variance of S are:

see A.160 equation

see A.162

Also, the mean resistance and its variance are:

Hence

Therefore c01-math-123 and from (1.21) and Appendix D

1.4.4 Safety Factors and Characteristic Values

The traditional deterministic measures of limit state violation, namely the factor of safety and the load factor, can be related directly to the probability pf of limit state violation. Analytically this is demonstrated most easily for the basic ‘one-resistance one-load-effect’ case, when R and S (or Q) are each normally distributed.

Consider a convenient simple safety measure sometimes referred to as the ‘central’ safety factor λ0 and defined as

1.23 equation

This definition does not accord with conventional usage, since generally some upper range value of applied load or stress is compared with some lower range value of strength of material. Such values might be termed ‘characteristic’ values, reflecting that in conventional usage (e.g. in design) the load or strength is described only by this value. Thus the characteristic yield strength of steel bars is the strength that most (say 95%) bars will exceed. There is a finite (but small) probability that some bars will have a lower strength.

For resistances, the design or ‘characteristic’ values are defined on the low side of the mean resistance (see Figure 1.12):

1.24 equation

where Rk is the characteristic resistance, μR the mean resistance, VR the coefficient of variation for R and kR a constant. This description is based on the Normal distribution. Rk is the value of